270      X   EXISTENCE THEOREMS
^ IMPLICIT FUNCTIONS
(102.1) (The implicit function theorem) Let E, F, G be three Banach
spaces yf a continuously differentiable mapping (Section 8.9) of an open subset
A ofE x F into G. Let (x0 , yQ) be a point of A such thatf(xQ , j0) = 0 and that
the partial derivative D2/(x0 , y0) be a linear homeomorphism of F onto G.
Then, there is an open neighborhood U0 of x0 inE such that, for every open
connected neighborhood U of x0 , contained in U0 , there is a unique continuous
mapping uofU into F such that u(xq) = y0 , (x, u(x)) e A andf(x, u(x)) = 0
for any xeU. Furthermore, u is continuously differentiable in U, and its
derivative is given by
(10.2.1.1)	u'(x) = -(D2/(x, W(x)))-1o(D1/(^ u(x))).
Let T0 be the linear homeomorphism D2/(#0 > ^o) of F onto G, T^1 the
inverse linear homeomorphism; write the relation f(x, y) = Q under the
equivalent form
(10.2.1.2)	y = y-T^-f(x.y)
and write g(x, y) the right-hand side of (1 0.2.1 .2). We are going to prove that
it is possible to apply (10.1.1) to the mapping
(X, /) -> #(*o + x', y0 + /) - y0
of E x F into F, in a sufficiently small neighborhood of (0, 0). As tq *o TQ = 1
by definition, we can write, for (x, y^ and (x, y2) in A,
9(x, yd ~ d(x, y2) = to l ' (D2f(x0 , j>0) • (y, - j2) - (/(x, yi) -/(x, y2)))-
Let e' > 0 be such that ellTV1 1| < i; as/is continuously differentiable in A, it
follows from (8.6.2) and (8.9.1) that there is a ball U0 (resp. V0) of center
jc0 (resp. j0) and radius a (resp. /?) in E (resp. F) such that, for x e U0 , y^ e
we have
\\f(x, yj -f(x, y2) - D2/(x0 , y0) • 0>i ~
whence H^,^)-^, ^K^ll^1!! • II* -y2\\ < ilbi -J'all for any
^ e U0 , j>! e V0 , y2 e V0 . On the other hand, g(x, yQ) - y0 = - tq 1 -f(x, y0) ;
as f(x0 , 70) = 0 and / is continuous, we can suppose e has been taken small
enough to have \\g(x9 y0) - y0\\ < j8/2 for x e U0 . We can then apply (1 0.1 .1 ),
which yields the existence and uniqueness of a mapping u of U0 into V0 , such
that/(x, u(x)) = 0 for every x e U0; as/(#0 , J0) = 0, this gives in particular
u(xq) = y0; finally u is continuous in U0 .
Next we prove that if U c U0 is a connected open neighborhood of jcq ,
u is the unique continuous mapping of U into F such that u(x0) = yQ ,
(x, u(x)) e A and f(x, u(x)) = 0. Let v be a second mapping verifying these